| 21. | If is a measurable function of the set to the reals ( including ), then we can write
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| 22. | To emphasize this dependency, if f : X \ to Y is a measurable function, we will write
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| 23. | The range of a lifting is always a set of measurable functions with the " separation property ".
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| 24. | The bounded measurable functions on " X " form a Banach space with respect to the supremum norm.
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| 25. | For instance, a "'real-valued measurable function "'is a function for which the preimage of each Borel set is measurable.
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| 26. | The Markov transition of the chain is given for any bounded measurable functions " f " by the formula
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| 27. | Real numbers in V [ G ] then correspond to Dedekind cuts of such functions, that is, measurable functions.
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| 28. | Every injective measurable function from a " standard " probability space to a " standard " measurable space is generating.
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| 29. | X \ colon \ Omega \ to E is a measurable function from the set of possible Measure-theoretic definition ).
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| 30. | The idea is to first establish the continuous functional calculus then pass to measurable functions via the Riesz-Markov representation theorem.
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